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New abstract Hardy spaces. (English) Zbl 1171.42012
Let \(\mathbb{X}\) be a set, \(d\) a quasi-distance on \(\mathbb{X}\) and \(\mu\) a Borel measure which satisfies the doubling property. For \(Q\) a ball and \(i\geq0\), set \(S_i(Q)=\{ x\in \mathbb{X }:2^i \leq 1+\frac {d(x,c(Q))}{r_Q}<2^{i+1}\}\), where \(r_Q\) is the radius of the ball \(Q\) and \(c(Q)\) its center. Let \(\mathcal L=\{B(x,r): x\in \mathbb{X}, r>0\}\) and \(\mathcal B=(B_Q)_{Q\in \mathcal L}\) be a collection of uniformly \(L^2\)-bounded linear operator, indexed by the collection \(\mathcal L\). Let \(\varepsilon>0\) be a fixed parameter. A function \(m\in L_{\text{loc}}^1\) is called an \(\epsilon\)-molecule associated to a ball \(Q\) if there exists a real function \(f_Q\) such that \(m=B_Q(f_Q)\) with for all \(i\geq0,\) \(\|f_Q\|_{2,S_i(Q)}\leq[\mu(2^iQ)]^{-\frac{1}{2}}2^{-\varepsilon i} \). If in addition \(\text{supp}(f_Q)\subset Q\), then \(m=B_Q(f_Q)\) is called an atom.
In this paper, the authors construct abstract Hardy spaces by a molecular (or atomic) decomposition and use the weakest assumptions to obtain good properties for these spaces. Mainly, the authors obtain a criterion for the continuity of an operator from the Hardy space into \(L^1(\mathbb X)\) and an interpolation result between the Hardy space and Lebesgue spaces. The authors also obtain some results on weighted norm inequalities and present partial results in order to understand a characterization of the duals of Hardy spaces.

MSC:
42B35 Function spaces arising in harmonic analysis
42B30 \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
43A99 Abstract harmonic analysis
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